On Spinor Varieties and Their Secants⋆

Varieties of sums of powers Dedicated to the memory of

The variety of sums of powers of a homogeneous polynomial of degree d in n variables is defined and investigated in some examples, old and new. These varieties are studied via apolarity and syzygies. Classical results (cf. [Sylvester 1851], [Hilbert 1888], [Dixon, Stuart 1906]) and some more recent results of Mukai (cf. [Mukai 1992]) are presented together with new results for the cases (n, d) ...

Varieties of Completely Decomposable Forms and Their Secants

This paper is devoted to the study of higher secant varieties of varieties of completely decomposable forms. The main goal is to develop methods to inductively verify the non-defectivity of such secant varieties. As an application of these methods, we will establish the existence of large families of non-defective secant varieties of “small” varieties of completely decomposable forms.

Secants of Abelian Varieties, Theta Functions, and Soliton Equations

On Secant Varieties of Compact Hermitian Symmetric Spaces

We show that the secant varieties of rank three compact Hermitian symmetric spaces in their minimal homogeneous embeddings are normal, with rational singularities. We show that their ideals are generated in degree three with one exception, the secant variety of the 21-dimensional spinor variety in P, whose ideal is generated in degree four. We also discuss the coordinate ring of secant varietie...

The Singularities of the 3-secant Curve Associated to a Space Curve

Let C be a curve in P3 over an algebraically closed field of characteristic zero. We assume that C is nonsingular and contains no plane component except possibly an irreducible conic. In [GP] one defines closed r-secant varieties to C, r £ N. These varieties are embedded in G, the Grassmannian of lines in P3. Denote by T the 3-secant variety (curve), and assume that the set of 4-secants is fini...